Hybrid Perturbation methods based on Statistical Time Series models

arXiv:1605.09058v1 [physics.space-ph] 29 May 2016

Hybrid Perturbation methods based on Statistical Time Series models Juan F´elix San-Juana,∗, Montserrat San-Mart´ına , Iv´an P´ereza , Rosario L´opezb a

Scientific Computing Group (GRUCACI), University of La Rioja, 26004 Logro˜ no, Spain Scientific Computing Group (GRUCACI), Center for Biomedical Research of La Rioja (CIBIR), 26006 Logro˜ no, Spain

b

Abstract In this work we present a new methodology for orbit propagation, the hybrid perturbation theory, based on the combination of an integration method and a prediction technique. The former, which can be a numerical, analytical or semianalytical theory, generates an initial approximation that contains some inaccuracies derived from the fact that, in order to simplify the expressions and subsequent computations, not all the involved forces are taken into account and only low-order terms are considered, not to mention the fact that mathematical models of perturbations not always reproduce physical phenomena with absolute precision. The prediction technique, which can be based on either statistical time series models or computational intelligence methods, is aimed at modelling and reproducing missing dynamics in the previously integrated approximation. This combination results in the precision improvement of conventional numerical, analytical and semianalytical theories for determining the position and velocity of any artificial satellite or space debris object. In order to validate this methodology, we present a family of three hybrid orbit propagators formed by the combination of three different orders of approximation of an analytical theory and a statistical time series model, and analyse their capability to process the effect produced by the flattening of the Earth. The three considered analytical components ∗

Corresponding author. Tel.: +34 941299440; fax: +34 941299460. Email addresses: [email protected] (Juan F´elix San-Juan), [email protected] (Montserrat San-Mart´ın), [email protected] (Iv´ an P´erez), [email protected] (Rosario L´opez)

Preprint submitted to Advances in Space Research

May 31, 2016

are the integration of the Kepler problem, a first-order and a second-order analytical theories, whereas the prediction technique is the same in the three cases, namely an additive Holt-Winters method. Keywords: Artificial Satellite Theory, Orbit propagator, hybrid perturbation method, time series 1. Introduction The equations of the perturbed motion of a satellite can be written as a set of 3 second-order or 6 first-order ordinary differential equations. The orbit propagation problem consists in computing the position and velocity of the satellite at a given final time tf , from the position and velocity at a given initial time t1 . Classically, the techniques used to solve this problem have been three. The first two methods are known as general and special perturbation techniques. General perturbation techniques are based on the analytical integration of the satellite’s equations of motion using perturbation theories (Deprit, 1969; Giacaglia, 1964; Hori, 1966, 1971; Krylov and Bogoliubov, 1943; Morrison, 1966). These techniques provide approximate analytical solutions (Aksnes, 1970; Brouwer, 1959; Hoots and Roehrich, 1980; Hoots and France, 1987; Kinoshita, 1977; Kozai, 1962; Lyddane, 1963) valid for any set of initial conditions. These solutions are explicit functions of time, physical parameters and integration constants, which are mainly characterized by retaining the essential behaviour of the motion. It is worth noting that most analytical theories currently in use only consider very basic models of external forces, because in some cases their corresponding analytical expressions can be too cumbersome. Furthermore, only low-order approximations are taken into account because analytical expansions for the higher-order solutions may become unmanageably long. Some of these theories can even implement truncated dynamic parameter expansions, so that their accuracy and computational efficiency are closely related to the initial assumptions. On the other hand, special perturbation methods (Berry and Healy, 2004; Kinoshita and Nakai, 1989; Long et al., 1989) refer to the accurate numerical integration of the equations of motion, including any external forces, even those in which analytical manipulations are complicated, which makes it necessary to use small steps in order to integrate the equations of motion. General perturbation methods produce more computationally efficient 2

propagators although are not as accurate as those developed using special perturbation techniques. Finally, the third approach is the semianalytical technique (Cefola et al., 2010; Liu and Alford, 1980; Neelon et al., 1998), which combines and takes advantage of the best characteristics of both the general and special perturbation techniques. This approach allows to include any external forces in the equations of motion, which are simplified using analytical techniques. Thus, the transformed equations of motion can be integrated numerically in a more efficient way by using longer integration steps. Current needs for Space Situational Awareness require improving orbit propagation of space objects in different ways, including the efficient shortterm propagation of catalogues of thousands of objects, the accurate verylong-term propagation needed for designing disposal strategies, the instant propagation of fragmentation models, or the propagation of uncertainties in observed orbits of Potentially Hazardous Objects, among others. Improvement in the models to be integrated constitutes a basic line of research, together with the use of advanced computer architectures based on parallel processing. Additional improvement can be achieved by combining both integrating and forecasting techniques, which we have called hybrid methods. In this work we present the hybrid perturbation theory, which may combine any kind of the aforementioned integration techniques with forecasting techniques based on statistical time series models (Chan and Ripley, 2012; Trapletti et al., 2015; San-Juan et al., 2012) or computational intelligence methods (P´erez et al., 2013). This combination allows for an increase in the accuracy of the numerical, analytical or semianalytical theories for predicting the position and velocity of any artificial satellite or space debris object, through the modelling of higher-order terms and other external forces not considered in those initial theories, as well as some physical effects not accurately modelled by the mathematical equations. The final goal of hybrid methodology is to complement the mathematical model of an orbiter dynamics, which is never a completely faithful representation of physical phenomena, with real dynamics provided by real observations, thus yielding a more accurate representation of real behaviour. As a first step in the process to eventually include unmodelled physical effects in the formulation of the problem, we start by considering a basic perturbation, J2 , and check the capability of the hybrid propagator to grasp its dynamics. In this process we simulate real observations by means of numerically generated ephemeris 3

through an 8th order Runge-Kutta method (Dormand and Prince, 1989). The aim of this paper is to develop a family of hybrid orbit propagators based on three different orders of approximation of an analytical theory, in order to model the effect produced by the flattening of the Earth so that this technique can be validated. These hybrid orbit propagators incorporate the integration of the Kepler problem in the first case, a first-order analytical theory in the second case and a second-order analytical theory in the last case as the integration techniques; the forecasting technique is an additive Holt-Winters method in the three cases. This paper is organized as follows. Section 2 describes the concept that underlies hybrid perturbation methodology. Section 3 outlines the secondorder analytical theory PPD that, together with its first-order and zero-order approximations, constitutes the base for the three hybrid propagators to be developed in the following sections. Section 4 describes the Holt-Winters method, an exponential smoothing technique used in this paper as the forecasting part of the hybrid propagators. In Section 5, the construction of the three hybrid propagators is detailed, paying special attention to the preliminary statistical analysis of control data, which is important in order to choose the most appropriate sampling rate for the time series to be processed. Results are analysed, and compared to the conventional analytical propagation results, for a set of 9 LEO satellites. Finally, Section 6 summarizes the study and remarks some interesting findings. 2. Hybrid perturbation methodology A hybrid perturbation theory is a methodology for determining an estimation of the position and velocity of any orbiter, which may be an artificial satellite or space debris object, at a final instant tf , in some set of canonical ˆ tf . or non-canonical variables, x In a first phase, an integration method I is needed in order to calculate a first approximation, xItf , from the position and velocity at an initial instant t1 , xt1 : xItf = I(tf , xt1 ).

(1)

This approximation can include some inaccuracies derived from the facts that, for the sake of manageability of the resulting expressions and affordability of the subsequent computations, not all the external forces are usually 4

taken into account in the physical model, and only low-order approximations are considered. Additional imprecision arises from the fact that mathematical models of perturbations not always depict real physical phenomena with high fidelity. The error of this approximation for any instant ti , εti , can be determined if the exact position and velocity xti is known, usually through a precise observation: εti = xti − xIti .

(2)

The second phase of the method requires the knowledge of xti for t1 , . . . , tT , with tT < tf , in order to build the time series of errors εt1 . . . εtT that contains the dynamics not present in the approximation generated during the first phase. The time elapsed between t1 and tT is defined as control period, εt1 . . . εtT as control data and T as the number of points in the control period. Then the goal is the modelling of such dynamics in order to be able to reproduce it; this task is accomplished by means of statistical techniques in time series analysis or computational intelligence methods. Once it has been done, an estimation of the error at the final instant tf , εˆtf , can be calculated, ˆ tf can be determined as: and consequently the desired value of x ˆ tf = xItf + εˆtf . x

(3)

It is worth noting that this methodology can be applied to any kind of integration methods regardless of the fact that in this work it has been applied to an analytical theory. For this case, Figure 1 shows the instants at which both the analytical expression and the statistical time series model have to be evaluated. 3. Second-order analytical theory PPD The main problem of the artificial satellite theory is defined as a Kepler problem perturbed by Earth’s oblateness. This model provides a first approximation to describe the motion of a low Earth orbiter. In this Section we develop a second-order closed-form analytical theory, based on Lie transforms, for the main problem, which we will use to derive the analytical part of the three hybrid orbit propagators to be analysed in Section 5. In the first case, a zero-order approximation, i.e. the Kepler solution, will be considered,

5

Analytical theory Propagation

t1 tT Statistical time series model estimation

tf

Figure 1: Evaluation of a hybrid propagator based on the combination of an analytical theory and a statistical time series model.

whereas a first-order and the complete second-order approximations will be taken into account for the remaining two cases. The main-problem dynamical system can be described in a cartesian coordinate system (x, X) by means of the Hamiltonian   α 2  z  1 µ H = (X · X) − 1 − J2 , (4) P2 2 r r r p where r = x2 + y 2 + z 2 is the geocentric distance, P2 represents the Legendre polynomial of degree 2, µ is the gravitational constant, α is the equatorial radius of the Earth, and J2 > 0 is a constant representing the shape of the Earth. In order to carry out a second-order analytical theory, the Hamiltonian (4) is rewritten in terms of Delaunay variables (l, g, h, L, G, H). This set of canonical action-angle variables can be directly related to the orbital elements through the following expressions: l = M, g = ω, h = Ω,

√ L = pµa, G = pµa(1 − e2 ), H = µa(1 − e2 ) cos i,

(5)

where M is the mean anomaly, ω the argument of the perigee, Ω the longitude of the ascending node, a the semi-major axis, e the eccentricity and i the inclination. Therefore, the transformed Hamiltonian in Delaunay variables yields 6

µ2  µ  α 2 H=− 2 − (1 − 3s2 sin2 (f + g)), (6) 2L 2r r where  = J2 is a small parameter, s = sin i and f is the true anomaly. Next, following the method described in (Deprit, 1969), three Lie transforms are applied in order to remove the long-period terms due to the argument of the perigee in the first place (Alfriend and Coffey, 1984; Deprit, 1981), and then the short-period terms due to the mean anomaly (Deprit, 1982). Finally, the transformed Hamiltonian yields, up to second order,
004 α 2 µ4 3α4 µ6  002 µ2 002 2 00 +  (3s − 2) −  5η + 36η + 35 s 2L002 4η 003 L006 128η 007 L0010

 +8 η 002 − 6η 00 − 10 s002 − 8 η 002 − 2η 00 − 5 , (7)

K = −

√ where η 00 = 1 − e002 . It is worth noting that K is independent of the variables l00 , g 00 and h00 , and thus Hamilton’s equations can be easily integrated by quadratures. The algebraic manipulations required to develop this analytical theory and its corresponding analytical orbit propagator program were built using a set of Mathematica packages called MathATESAT (San-Juan et al., 2011), which is a reimplementation of the ATESAT (San-Juan, 1994, 1998). The acronym PPD makes reference to the sequence of Lie transforms used to carry out this analytical theory; in this case, the involved transforms are the elimination of the Parallax, the elimination of the Perigee and the Delaunay normalization. 4. Time series forecasting using exponential smoothing methods Exponential smoothing methods are forecasting algorithms for time series. Their main advantages are their ease of application, speed and reduced computational burden. Predictions generated by these methods are based on previously collected data, giving higher importance to more-recent observations. These methods assume a time series is the combination of three components: the trend or long-term variation, the seasonal component, which represents periodic oscillations that repeat at constant intervals, and the irregular or non-predictable component. There are two main procedures for combining these components, depending on the cyclic behaviour with respect 7

to the trend: the additive and the multiplicative compositions. In the additive case, the series shows stable cyclic fluctuations, independently of the increase in the series level. On the contrary, the multiplicative model implies a change in the amplitude of the seasonal oscillations as the series trend varies. It is worth noting that a multiplicative model can be converted into an additive one through a Box-Cox transformation. In mathematical terms, a time series, εt , can be decomposed into trend, µt , seasonal variation, st , and irregular or non-predictable component, νt . In an additive model, these components combine in the following manner: ε t = µ t + s t + νt .

(8)

In particular, the Holt-Winters method (Winters, 1960) combines a linear trend together with a periodic behaviour. In this method, the trend can be expressed as: µt = a + bt,

(9)

where a and b represent the level and slope of the series, respectively. This method predicts the series value at time t according to the following recursive procedure εˆt = At−1 + Bt−1 + St−s ,

(10)

that is, the addition of level, At−1 , and slope, Bt−1 , at the previous instant, plus seasonal variation, St−s , s epochs before, thus being s the period of such seasonal variation. The corresponding algorithm updates, at every epoch, the level, the slope of the trend and the values of the seasonal factors, by means of three equations. The first equation determines the series level at epoch t, At , as the weighted average of the deseasonalized series value at the same instant t and the non-seasonal prediction at the previous epoch, that is, At = α(εt − St−s ) + (1 − α)(At−1 + Bt−1 ),

(11)

where α is a constant, named smoothing parameter, with values in the interval [0, 1]. With the equation Bt = β(At − At−1 ) + (1 − β)Bt−1 8

(12)

the slope can be estimated as the weighted average of the slope at the previous epoch and its corresponding level change. The smoothing parameter β can have values in the interval [0, 1]. The last equation determines the seasonal component at epoch t, St , as the weighted average of the detrended series and the seasonal value at the equivalent epoch in the previous period, St = γ(εt − At ) + (1 − γ)St−s ,

(13)

where γ is another smoothing parameter which can also have values in the interval [0, 1]. The smoothing parameters α, β and γ are decisive in the estimation process. Parameter α controls the smoothing of the level equation, so that low values give more importance to historical data, whereas high values weight recent observations. Parameter β modifies the slope estimation in such a way that a value close to 0 gives more importance to trend, whereas a value near 1 weights level changes. Finally, γ controls the smoothing of the seasonal component, so that high values lead to predictions more sensitive to the series variations. Algorithm 1 implements the Holt-Winters method; its inputs are the number of data per revolution, s, the number of revolutions for which precise observations are available, c, the epoch number, starting after the last available precise observation, for which the series value has to be predicted, h, and the error series {εt }Tt=1 values with T = s × c. The algorithm is designed to produce εˆT +h|T as the output, which represents the time-series forecast at the final instant tf = T + h, based on the time-series value at the end of the control period T . The first step consists in estimating the initial values A0 , B0 , S−s+1 , . . ., S−1 and S0 by means of a heuristic method in which, in the first place, a classical additive decomposition into trend and seasonal variation over the two first revolutions of the satellite is performed. By doing so, the initial values of the seasonal component, S−s+1 , . . ., S−1 and S0 , are obtained, whereas the linear regression coefficients over the trend lead to the initial values of level and slope, A0 and B0 . Once these values have been obtained, the next step can be undertaken, in order to apply the recursive equations which allow for the calculation of the values of the components At , Bt and St , as well as the single-step error prediction εˆt for control data, i.e. for t = 1, . . . , T (lines 2–7). These values will remain as functions of the smoothing parameters α, 9

Algorithm 1 Holt-Winters Require: s, c, h and {εt }Tt=1 Ensure: εˆT +h|T 1: Estimate the values of A0 , B0 , S−s+1 , . . . , S−1 , S0 2: for t = 1; t ≤ T ; t = t + 1 do 3: At = α(εt − St−s ) + (1 − α)(At−1 + Bt−1 ) 4: Bt = β(At − At−1 ) + (1 − β)Bt−1 5: St = γ(εt − At ) + (1 − γ)St−s 6: εˆt = At−1 + Bt−1 + St−s 7: end for 8: Select error measure ∈ {MSE,MAE,MAPE} and obtain it as a function of the smoothing parameters 9: Obtain the smoothing parameters that minimize error measure using the L-BFGS-B method 10: Calculate AT , BT , ST −s+1 , . . . , ST −1 , ST for the optimal smoothing parameters 11: εˆT +h|T = AT + hBT + ST −s+1+h mod s 12: return εˆT +h|T β and γ. In the next step, one of the error functions is selected T 1X MSE = (εt − εˆt )2 , T i=1 T 1X |εt − εˆt |, T i=1 T 1 X εt − εˆt 100, MAPE = T i=1 εt

MAE =

(14)

and its value is determined as a function of the smoothing parameters. Next, the values of the smoothing parameters that minimize the chosen error function have to be determined. As it is not easy to minimize the error functions (14) analitycally, a numerical optimization method is necessary. The limited memory algorithm L-BFGS-B, which is one of the most usual ones, has been chosen for that purpose. The L-BFGS-B method (Byrd et al., 1995), which is a variation of the BFGS method (Broyden, 1970; Fletcher, 1970; 10

Goldfarb, 1970; Shanno, 1970), named after its creators Broyden, Fletcher, Goldfarb and Shanno, is a quasi-Newton limited memory algorithm that allows optimization with restrictions, thus permitting to impose limitations on smoothing parameters. With the optimal smoothing parameters, the level and slope values for the last control data (AT and BT ) are calculated, as well as the seasonal component values for the last revolution in control data (ST −s+1 , . . . , ST −1 , ST ). With these data, it is possible to predict the value of the series h epochs ahead, εˆT +h|T (line 11). 5. Validation of the methodology The proposed methodology is applied to the main problem of the artificial satellite theory so as to model the effect produced by the flattening of the Earth, which corresponds to the J2 term of Earth’s gravitational potential. Three analytical orbit propagator programs (AOPP) are used to conduct this study. They are derived from a second-order closed-form analytical theory based on Lie transforms, which has been briefly described in Section 3. The first AOPP is PPD0, a propagator derived from the zero-order analytical theory, in which only the part corresponding to Kepler’s problem has been taken into account. PPD1 is the second propagator; it implements the firstorder analytical theory, i.e. the first-order J2 approximation. Finally, PPD2 implements the second-order analytical theory, i.e. the full second-order J2 approximation. From each of these AOPPs, a hybrid analytical orbit propagator program (HAOPP) has been developed. In these HAOPPs, statistical time series analysis has been applied to forecast the effects not taken into account in their corresponding initial AOPPs. The propagator HPPD0 will be used to demonstrate the capability of this methodology to model the full J2 effect. It is worth noting that this perturbation is not included in the initial propagator PPD0 at all. On the other hand, the propagators HPPD1 and HPPD2 will be used to explore the capability of this methodology to model the error introduced by the analytical approximations, O(J22 ) and O(J23 ) respectively. In this work, the additive Holt-Winters method will be used for forecasting the effects not taken into account in the initial AOPPs. Finally, in order to compare and contrast the performance of the HAOPPs, several tests with numerically-simulated initial conditions corresponding to

11

LEO orbits will be performed. The error measure to be considered will be the distance error over a prediction horizon of 30 days. 5.1. Data preprocessing This methodology starts by choosing the set of variables that will be used for modelling purposes in the forecasting part of the hybrid propagator. In this work, Delaunay variables have been chosen, although other sets of variables can also be used. After that, in order to build the new propagator, two sets of values corresponding to the same satellite are necessary during the control period. The first consists of accurate values, obtained through the numerical integration of the original problem (4) by using a high-order Runge-Kutta method (Dormand and Prince, 1989), which are considered as actual values from precise observations. The second is obtained by applying the initial integrating part of the hybrid propagator; it contains approximate values which do not include, either in whole or in part, the effect that we want to model in the forecasting part. It is worth noting that the control data should include an amount of values which is enough to identify any pattern that we expect the forecasting part to model and reproduce. Then, subtracting both data sets for each variable the error time series l g h L G H (εt , εt , εt , εt , εt , εt ) are obtained. After this operation, the angular-variable time series εlt , εgt and εht may include some outliers that differ from the rest of values in a quantity multiple of 2π. Such differences correspond to complete spins and, although they have no effect on trigonometrical calculations, for time series analysis they represent abrupt discontinuities in values that are actually very close. Next, by adding or subtracting complete spins (2π), their values can be homogenized to the interval (−π, π], thus avoiding this and other problems related to the periodic behaviour of these series. The time series εH t is always 0 for the problem considered here, which means that the pure analytical theory is able to determine H values accurately. Therefore, forecasting of this time series is not necessary. For each of the remaining time series, an univariate Holt-Winters model will be developed from a preliminary analysis so as to forecast its future values. This analysis includes the study of the sequence graphics, periodograms and autocorrelation functions (ACF). These graphics can help reveal the most important characteristics of the series, such as trend, stationarity, atypical values, etc. Table 1 shows the orbital elements for nine fictitious satellites used for testing the HAOPPs. These initial conditions correspond to LEO orbits and

12

all of them have been chosen to avoid the intrinsic singularities present in Delaunay variables. Table 1: Satellites used for validation studies of hybrid methodology.

Id a (km) 1 7228 2 7872 3 7612 4 7674 5 7064 6 7087 7 6992 8 7269 9 7128

e Inclination (deg) 0.0631 49 0.1380 144 0.1132 102 0.1124 68 0.0323 62 0.0504 73 0.0268 29 0.0713 66 0.0499 66

Period (min) 101.926 115.936 110.156 111.505 98.477 98.958 96.975 102.795 99.818

5.2. Modelling the full J2 effect Satellite 1 will be used in this subsection to illustrate this methodology. It is necessary to start mentioning that in the case of the Kepler problem, the orbital elements are constant over time (with the unique exception of the mean anomaly M , which varies between 0 and 2π during each revolution of the satellite). However, adding the J2 effect to the Kepler problem, i.e. the main problem, produces significant effects on the orbital elements, which include secular, long and short period variations. Table 2 shows the distance error between PPD0 and the numerical integration of the main problem at different instants. The distance error grows up to a maximum value of about 14500 kilometers, which represents approximately the distance between perigee and apogee. These values will be used to evaluate the improvement introduced by the hybrid propagators. The first step in the preliminary analysis consists in plotting the time series εlt , εgt , εht , εLt and εG t . Initially, values have been generated every 10 minutes. Figures 2 and 3 only show the sequence plots, periodograms and autocorrelation functions of the series εlt , εht and εLt , because the behaviour l L of εgt and εG t is approximately the same as εt and εt respectively. First, we begin analysing the sequence plots. The series εlt has a linear increasing trend and a periodic behaviour with a period of approximately 13

Table 2: Maximum distance error between PPD0 and the numerical integration of the main problem for satellite 1 over different propagation spans.

Time 17 hours 1 day 2 days 7 days 30 days

Distance error (km) 864.80 1202.24 2408.31 7894.81 14506.50

Εlt HradL

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33.33 minutes. On the other hand, the trend of εht is linear and decreasing, but its periodic behaviour is difficult to discover because it is hidden by its trend, so other statistical tools are needed for its detection. Finally, the trend of εLt is not significant, so it can be considered constant; however, this series shows seasonal fluctuations, the first with a period of almost 50 minutes, whereas a repetitive pattern of alternate oscillations can be clearly observed approximately every 100 minutes. Then, the periodograms and the autocorrelation functions are analysed. It is worth noting that these functions can only be applied to stationary series so, in the first place, it will be necessary to differentiate the series εlt and εht so as to remove their trends. The periodogram of ∇εlt (Figure 3) shows that the most significant frequency is close to 0.6π, that is, a periodic behaviour with an approximate period of 33.33 minutes, whereas series ∇εht and εLt have their main frequency near 0.4π, which corresponds to a period of almost 50 minutes. Finally, from the analysis of the autocorrelation functions of series ∇εlt , ∇εht and εLt , it can be observed that, despite having high correlation in several delays, the strongest one corresponds to lag 10, which implies a close relationship each 10 points (approximately each 100 minutes). This preliminary analysis allows us to conclude that, although, in principle, it might seem that there are three main periodicities of approximate periods 33.33, 50 and 100 minutes, in reality, the most remarkable is the last one, which corresponds approximately to the Keplerian period of satellite 1, 101.926 minutes. Then, after estimating the initial values A0 , B0 , S−s+1 , . . . , S−s and S0 by means of the heuristic method described in Section 4, the next step consists in the identification of the optimal values for the smoothing parameters α, β and γ of the Holt-Winters method that minimize the distance error. As shown in Algorithm 1, their values are obtained by applying the L-BFGS-B algorithm to one of the error functions (14), MSE in this case. Finally, this model is analysed in order to experimentally determine the best amount of control data by choosing the number of both satellite revolutions and points per revolution which minimize the distance error over 30 days, taking into account that we are considering here the Keplerian period as the revolution period. We proceed by fixing the number of revolutions in the first place, with the aim of determining the optimal amount of points per revolution. Once it has been done, the best number of revolutions can be determined. Several configurations have been tested, keeping in mind as 16

general guidelines that the control period has to be long enough so as to contain any dynamics to be modelled, and the sampling rate needs to be high enough so as to capture the highest frequency that the hybrid propagator is designed to model and reproduce. This analysis leads us to consider 10 satellite revolutions and 12 points per revolution as the best choice to constitute the control data for all the studied satellites, which represents a time span of approximately 17 hours in the case of satellite 1. More details about this analysis can be found in San-Mart´ın (2014), although further research on this issue is being conducted in order to draw general conclusions regarding the optimal configuration of the control period. Table 3 shows the distance error between HPPD0 and the numerical integration of the main problem at different instants. As can be seen, the distance error, even after a 30-day propagation, is lower than the pure analytical propagator PPD0 distance error only after 17 hours, i.e. the control period. Table 3: Maximum distance error between HPPD0 and the numerical integration of the main problem for satellite 1 over different propagation spans.

Time 17 hours 1 day 2 days 7 days 30 days

Distance error (km) 2.69 2.85 3.10 10.83 13.79

Figure 4 shows how the differences between the orbital elements of both HPPD0 and the numerical integration of the main problem evolve during 2 days for satellite 1. Finally, Figure 5 compares the distance error for both the zero-order hybrid propagator HPPD0 and the first-order pure analytical propagator PPD1. The error is lower for the analytical propagator during the first 25 days, as could be expected from a higher-order approximation, although both errors become similar as the last part of the 30-day studied period is reached.

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30

t HdaysL

Figure 5: Distance error of the zero-order hybrid propagator HPPD0 (blue line) and the first-order pure analytical propagator PPD1 (red line) for satellite 1.

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5.3. Improving the analytical approximation In this subsection we study the hybrid propagators based on the first-order and second-order analytical propagators, PPD1 and PPD2 respectively. All the analysed hybrid propagators improve the accuracy of the analytical theory they are based on; nevertheless, preliminary analysis leads us to conclude that, similarly to the zero-order propagator, optimum results are achieved when the exact revolution period is taken into account, in such a way that the data sampling rate is chosen in order to allow for a complete number of samples per satellite revolution. Furthermore, the fact that not only periodograms and autocorrelation functions, but also the smoothing parameters to be obtained, are quite similar for the three studied orders of approximation, indicates that the error series εlt , εgt , εht , εLt and εG t maintain the same characteristics to be estimated, albeit with decreasing magnitudes for higher-order models, and consequently must be processed and forecasted by means of similar models. According to what has been expounded, hybrid propagators that consider M SE as the error function to be minimized, and take exactly 12 samples, at a regular rate, for each of the 10 revolutions that constitute the control period, are again the most accurate, leading to the best results for short, medium and long-term estimation. Figure 6 plots the differences in the evolution of the orbital elements of both the first-order hybrid propagator HPPD1 and the accurate numerical integration of the main problem for satellite 1. Table 4 shows the predictive capability of both the analytical and associated optimum hybrid propagators. It is worth noting that, after a 30-day propagation, distance error can be reduced by a factor of 20 in the case of first-order theory, or nearly 70 for second-order theory. Figure 7 plots the evolution of the distance error for the first-order hybrid propagator HPPD1 during a 30-day time span. It should be noted that the error remains quite low, just about 10 metres, for more than 5 days. 5.4. Hybrid propagators for the complete set of satellites The importance of acquiring the seasonal variations originated by the satellite revolution period has been verified for satellite 1 in previous subsections. Therefore, for the rest of the satellites, data will also be considered with sampling rates adapted to each satellite period, in such a way that a complete number of samples per revolution is always guaranteed. Apart from that, sequence graphics, periodograms and autocorrelation functions show 19

4 e‰10-6

a‰10-3 HkmL

0.5 0

-6

0

-1 0

0.7

1 t HdaysL

2

0

0.7

1 t HdaysL

2

0

0.7

1 t HdaysL

2

0

0.7

1 t HdaysL

2

Ω‰10-4 HdegreeL

i‰10-5 HdegreeL

4 1 0

-8

-2 0

0.7

1 t HdaysL

2

6

8 M‰10-4 HdegreeL

W‰10-5 HdegreeL

0

0

0 -4

-4 0

0.7

1 t HdaysL

2

Figure 6: Evolution of the orbital-element differences between HPPD1 and the numerical integration of the main problem for satellite 1.

Table 4: Distance error of the pure analytical and the hybrid propagators for both first and second-order theories (satellite 1).

Time 17 hours 1 day 2 days 7 days 30 days

PPD1(km) 0.2758 0.4037 0.8223 2.9175 12.5706

HPPD1(km) PPD2(km) HPPD2(km) 0.0008 0.0007 4.5 × 10−6 0.0015 0.0010 8.1 × 10−6 0.0061 0.0020 1.9 × 10−5 0.0548 0.0070 4.2 × 10−5 0.6462 0.0290 4.2 × 10−4

20

Distance error HkmL

0.6

0

0

10

20

30

t HdaysL

Figure 7: Distance error of the first-order hybrid propagator HPPD1 for satellite 1

that, in addition to the revolution-time periodicity, other two periodicities with magnitudes half and third the same satellite revolution time also exist. For those reasons, and with the aim of acquiring such periodic behaviour, it would be desirable to choose a number of samples per revolution which is multiple of both 2 and 3. Taking into account that it is never advisable to consider an amount of data either too high or too low, it is concluded that 12 samples per satellite revolution is an appropriate value as sampling rate also for the remaining 8 satellites. Figure 8 and Table 5 show the distance error of the pure analytical and the hybrid propagators for both zero and first-order theories after a 30-day propagation for the complete set of satellites analysed in this work. Taking into account that Figure 8 has been plotted with logarithmic scale, it should be noted that, in general, the distance error is between 2 and 3 orders of magnitude lower for the hybrid propagator than for the pure analytical propagator in the case of the zero-order theory, i.e. HPPD0 versus PPD0. This hybrid propagator HPPD0 has distance errors between only 0 and 1 orders of magnitude higher than the first-order analytical propagator PPD1, which is one order of approximation higher, although in the case of satellite 7 the hybrid propagator is even more accurate than the superior analytical propagator. When the two first-order propagators are compared, it is found that the hybrid HPPD1 is between 1 and 2 orders of magnitude more accurate than the analytical PPD1. 6. Conclusion In this work, we have presented a new approach called hybrid perturbation theory. The proposed methodology, which combines an integration method and a prediction technique, has been illustrated through the combination of 21

1000 PPD0 HPPD0

100

10 1

2

3

4

5

6

7

8

Distance error, log scale HkmL

Distance error, log scale HkmL

10 000

9

100

10 HPPD0 PPD1

1

HPPD1

0.1 1

2

3

Satellite Id

4

5

6

7

8

9

Satellite Id

Figure 8: Distance error of the pure analytical and the hybrid propagators for both zero and first-order theories after a 30-day propagation (satellites 1-9).

Table 5: Distance error of the pure analytical and the hybrid propagators for both zero and first-order theories after a 30-day propagation (satellites 1-9).

Id

PPD0 (km)

HPPD0 (km)

PPD1(km)

HPPD1 (km)

1 2 3 4 5 6 7 8 9

14506.5 16183.7 15987.4 15922.4 14292.2 14456.9 14012.8 14882.6 14489.9

13.792 49.136 146.465 107.905 23.774 128.633 27.992 84.369 114.199

12.6 27.3 12.5 2.3 6.9 8.7 68.6 3.4 3.2

0.634 0.918 0.101 0.106 0.083 0.058 0.255 0.096 0.115

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a simplified general perturbation theory and a statistical time series model based on an additive Holt-Winters method. The hybrid propagators that have been developed have proven an increase in the accuracy of the analytical theory for predicting the position and velocity of the studied orbiters, as well as modelling higher-order terms and other external forces not considered in the analytical theory. It has been found that the effect of considering a complete number of samples per revolution in hybrid propagators varies depending on the order of the underlying analytical theory, and thus on its margin for improvement. In the case of hybrid propagators based on the zero-order analytical theory, a dramatic reduction in distance error is achieved. In contrast, the secondorder hybrid propagator, whose margin for improvement is very reduced, only reaches a slight increase in accuracy when the sample rate is such that a complete number of values fits into a satellite revolution. Another remarkable conclusion is that similar smoothing parameters are obtained for hybrid propagators based on a certain analytical theory with different orders of approximation, which implies that the error time series to be modelled maintain the same characteristics, even though their magnitude can vary. Acknowledgments This work has been funded by the Spanish Finance and Competitiveness Ministry under Project ESP2014-57071-R. The authors would like to thank the reviewers for their valuable suggestions. References Aksnes, K., November 1970. A second-order artificial satellite theory based on an intermediate orbit. The Astronomical Journal 75 (9), 1066–1076. Alfriend, K. T., Coffey, S. L., February 1984. Elimination of the perigee in the satellite problem. Celestial Mechanics 32 (2), 163–172. Berry, M. M., Healy, L. M., July 2004. Implementation of Gauss-Jackson integration for orbit propagation. The Journal of the Astronautical Sciences 52 (3), 331–357. Brouwer, D., November 1959. Solution of the problem of artificial satellite theory without drag. The Astronomical Journal 64 (1274), 378–397. 23

Broyden, C. G., 1970. The convergence of a class of double-rank minimization algorithms. 1. General considerations. IMA Journal of Applied Mathematics 6 (1), 76–90. Byrd, R. H., Lu, P., Nocedal, J., Zhu, C., 1995. A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing 16 (5), 1190–1208. Cefola, P. J., Phillion, D., Kim, K. S., 2010. Improving access to the semianalytical satellite theory. Advances in the Astronautical Sciences 135, 46 pages, paper AAS 09-341. Chan, K. S., Ripley, B., 2012. TSA: time series analysis. R Foundation for Statistical Computing, R package version 1.01. URL http://CRAN.R-project.org/package=TSA Deprit, A., March 1969. Canonical transformations depending on a small parameter. Celestial Mechanics 1 (1), 12–30. Deprit, A., June 1981. The elimination of the parallax in satellite theory. Celestial Mechanics 24 (2), 111–153. Deprit, A., January 1982. Delaunay normalisations. Celestial Mechanics 26 (1), 9–21. Dormand, J. R., Prince, P. J., 1989. Practical Runge-Kutta processes. SIAM Journal on Scientific and Statistical Computing 10 (5), 977–989. Fletcher, R., August 1970. A new approach to variable metric algorithms. The Computer Journal 13 (3), 317–322. Giacaglia, G. E. O., June 1964. Notes on von Zeipel’s method. Tech. Rep. X-547-64-161, Goddard Space Flight Center, Greenbelt, MD, USA. Goldfarb, D., 1970. A family of variable-metric methods derived by variational means. Mathematics of Computation 24, 23–26. Hoots, F. R., France, R. G., March 1987. An analytic satellite theory using gravity and a dynamic atmosphere. Celestial Mechanics 40 (1), 1–18.

24

Hoots, F. R., Roehrich, R. L., 1980. Models for propagation of the NORAD element sets. Spacetrack Report #3, U.S. Air Force Aerospace Defense Command, Colorado Springs, CO, USA. Hori, G.-i., 1966. Theory of general perturbations with unspecified canonical variables. Publications of the Astronomical Society of Japan 18 (4), 287– 296. Hori, G.-i., 1971. Theory of general perturbations for non-canonical systems. Publications of the Astronomical Society of Japan 23, 567–587. Kinoshita, H., July 1977. Third-order solution of an artificial-satellite theory. Special Report no. 379, Smithsonian Astrophysical Observatory, Cambridge, MA, USA. Kinoshita, H., Nakai, H., 1989. Numerical integration methods in dynamical Astronomy. Celestial Mechanics 45 (1), 231–244. Kozai, Y., September 1962. Second-order solution of artificial satellite theory without air drag. The Astronomical Journal 67 (7), 446–461. Krylov, N. M., Bogoliubov, N. N., 1943. Introduction to Non-linear Mechanics. Princeton University Press, Princeton, NJ, USA. Liu, J. J. F., Alford, R. L., July 1980. Semianalytic theory for a close-Earth artificial satellite. Journal of Guidance, Control, and Dynamics 3 (4), 304– 311. Long, A. C., Cappellari, J. O., Velez, C. E., Fuchs, A. J., July 1989. Goddard Trajectory Determination System (GTDS) mathematical theory (revision 1). Tech. Rep. CSC/TR-89/6001, Computer Sciences Corporation. Lyddane, R. H., October 1963. Small eccentricities or inclinations in the Brouwer theory of the artificial satellite. The Astronomical Journal 68 (8), 555–558. Morrison, J. A., January 1966. Generalized method of averaging and the von Zeipel method. In: Duncombe, R. L., Szebehely, V. G. (Eds.), Methods in Astrodynamics and Celestial Mechanics. Vol. 17 of Progress in Astronautics and Aeronautics. American Institute of Aeronautics and Astronautics, pp. 117–138. 25

Neelon, Jr., J. G., Cefola, P. J., Proulx, R. J., 1998. Current development of the Draper Semianalytical Satellite Theory standalone orbit propagator package. Advances in the Astronautical Sciences 97, 2037–2052, paper AAS 97-731. P´erez, I., San-Juan, J. F., San-Mart´ın, M., L´opez-Ochoa, L. M., 2013. Application of computational intelligence in order to develop hybrid orbit propagation methods. Mathematical Problems in Engineering 2013, 11 pages, article ID 631628. San-Juan, J. F., 1994. ATESAT: automatization of theories and ephemeris in the artificial satellite problem. Tech. Rep. CT/TI/MS/MN/94-250, Centre ´ National d’Etudes Spatiales (CNES), Toulouse, France. San-Juan, J. F., 1998. ATESAT: review and improvements. Study of a family of analytical models of the artificial satellite generated by ATESAT and their numerical validation versus PSIMU and MSLIB. Tech. Rep. ´ DGA/T/TI/MS/MN/97-258, Centre National d’Etudes Spatiales (CNES), Toulouse, France. San-Juan, J. F., L´opez, L. M., L´opez, R., 2011. MathATESAT: A symbolicnumeric environment in Astrodynamics and Celestial Mechanics. Lecture Notes in Computer Science 6783 (2), 436–449. San-Juan, J. F., San-Mart´ın, M., P´erez, I., 2012. An economic hybrid J2 analytical orbit propagator program based on SARIMA models. Mathematical Problems in Engineering 2012, 15 pages, article ID 207381. San-Mart´ın, M., 2014. M´etodos de propagaci´on h´ıbridos aplicados al problema del sat´elite artificial. T´ecnicas de suavizado exponencial. Ph.D. thesis, University of La Rioja, Spain. Shanno, D. F., July 1970. Conditioning of quasi-Newton methods for function minimization. Mathematics of Computation 24 (111), 647–656. Trapletti, A., Hornik, K., LeBaron, B., 2015. tseries: time series analysis and computational finance. R Foundation for Statistical Computing, R package version 0.10-34. URL http://CRAN.R-project.org/package=tseries

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Winters, P. R., 1960. Forecasting sales by exponentially weighted moving averages. Management Science 6 (3), 324–342.

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